Adapting Mathematics Education to the Needs of ICT Haapasalo, L. (2007). Adapting mathematics education to the needs of ICT. The Electronic Journal of Mathematics and Technology, 1(1), Reporter: Lee Chun-Yi Advisor: Chen Ming-Puu

Abstract Looking the relationship between technology and mathematics education from five perspectives  MODEM framework  Eight main motives and activities

Introduction Technology-based (or ICT-based) mathematics education has expanded to include the following solutions:  computer algebra systems (CAS), dynamical geometry (DGS), and dynamical statistics (DSS);  spreadsheets, drawing programs, and other versatile tools for mathematical modelling;  online databases of available software, instruction, research, statistics, history, etc.;  online communication in all of its synchronous and asynchronous forms;  new kinds of environments to read, write and publish, including tools for support;  tools for utilizing of the world-wide web: search engines, etc.;  online experiments and simulations in diverse forms of digital educational content  online libraries containing books, learning objects, other teaching materials, digital portfolios, etc.  learning management systems (LMS), which are used to manage students and course materials;  virtual worlds in the form of three-dimensional immersive environments offering, for example, shared exhibitions or other forms of collaborative functionality.

Introduction Kadijevich (2004) points out four areas that have been neglected in research in mathematics education:  promoting the human face of mathematics; relating procedural and conceptual mathematical knowledge;  utilizing mathematical modelling in a humanistic, technology-supported way;  promoting technology-based learning through applications and modelling, multimedia design, and on- line collaboration.

Introduction Haapasalo and Siekkinen (2005) find support for the following hypotheses:  Technology can enhance learning skills (metacognitions) among teachers and students;  it is reasonable to utilize minimalist instruction especially when technology concerns;  technology can shift learning from the classroom into free time; technology-based learning can benefit from the ‘learning by design’ principle;  and that the most appropriate way to implement technology in teacher training is to use it as a solid part of knowledge structure and of student pedagogical thinking.

Introduction Modern technology can maintain and promote: (1) Links between conceptual and procedural knowledge, (2) Metacognitions and problem-solving skills, (3) Sustainable components of mathematics making, (4) Interplay between systematic approaches and minimalist instruction, and (5) Learning by design. By using examples of empirical studies I will try to emphasize that these aspects might very often be related to technology in more natural way outside the classroom than within institutional teaching.

Links between conceptual and procedural knowledge Procedural knowledge denotes dynamic and successful use of specific rules, algorithms or procedures within relevant representational forms. This usually requires not only knowledge of the objects being used, but also knowledge of the format and syntax required for the representational system(s) expressing them. Conceptual knowledge denotes knowledge of particular networks and a skilful “drive” along them. The elements of these networks can be concepts, rules (algorithms, procedures, etc.), and even problems (a solved problem may introduce a new concept or rule) given in various representational forms.

Links between conceptual and procedural knowledge MODEM Project (

Links between conceptual and procedural knowledge Developmental approach  Genetic view: procedural knowledge is necessary for the conceptual.  Simultaneous activation view: procedural knowledge is necessary and sufficient for conceptual knowledge. Education approach  Dynamic interaction view: conceptual knowledge is necessary for the procedural.  Simultaneous activation view: conceptual knowledge is necessary and sufficient for procedural knowledge.

Links between conceptual and procedural knowledge Relating different representations can not only support the development of conceptual knowledge (cf. Papert 1987), but also relate procedural and conceptual knowledge (cf. Haapasalo 2003, Kadijevich & Haapasalo 2001). To coordinate the process and object features of mathematical knowledge, multiple forms of representation are to be utilized and connected, especially with the aid of modern technological tools. The use of these tools should not reinforce a strictly hierarchical nature of mathematical knowledge but rather promote its quality of a flexible network (Kadijevich 2006).

Metacognitions and Problem-Solving Skills Students very often use modern technology in very sophisticated way outside the classroom. Kidware aquarium simulation program  Haapasalo & Siekkinen (2005) report that after using these programs during two years, children’s metacognitions increased, whilst entertainment- relatedness decreased.

Metacognitions and Problem-Solving Skills

Sustainable Components of Mathematics Making Zimmermann’s (2003) long-term study of the history of mathematics reveals eight main motives and activities, which proved to lead very often to new mathematical results at different times and in different cultures for more than 5000 years. We (Eronen & Haapasalo 2006) took this network of activities illustrated in Figure 5 as an element in our theoretical framework for the structuring of learning environments and for analyzing student’s cognitive and affective variables. Within these activities ClassPad study focuses on “changing representation” which is not only a powerful thinking tool to enhance problem solving processes but it might also promote links between procedural and conceptual knowledge.

Sustainable Components of Mathematics Making Affective resultsSelf-confidenceComputer’s role

Sustainable Components of Mathematics Making Modern technology can not only promote those eight dimensions of mathematics making but it can also revitalize beautiful mathematical ideas, which have been developed by great mathematicians through centuries.  The rich concept curve served as an assembler of the isolated parts of mathematics: Geometry, Algebra, Trigonometry, Analytical Geometry, Calculus. Among the huge amount of material on the Internet developed widely, there might be useful for any educator who can use search engines in sophisticated way and has proper conceptual mathematical knowledge.  This opportunity was utilized by Kadijevich & Haapasalo (2004) when students designed their own hypermedia utilizing Internet.

Interplay between Systematic Approaches and Minimalist Instruction Minimalist instruction  Carroll observed that learners often tend to “jump the gun”.  They avoid careful planning, resist detailed systems of instructional steps, tend to be subject to learning interference from similar tasks, and have difficulty recognizing, diagnosing, and recovering from their errors.

Interplay between Systematic Approaches and Minimalist Instruction Characteristics of minimalist instruction (cf. Lambrecht 1999):  Specific content and outcomes cannot be pre-specified, although a core knowledge domain may be specified;  Learning is modelled and coached for students with unscripted teacher responses;  Learning goals are determined from authentic tasks stressing doing and exploring;  Errors are not avoided but used for instruction;  Learners construct multiple perspectives or solutions through discussion and collaboration;  Learning focuses on the process of knowledge construction and development of reflexive  awareness of that process;  Criterion for success is the transfer of learning and a change in students’ action potential;  The assessment is ongoing and based on learner needs.

Interplay between Systematic Approaches and Minimalist Instruction Systematic Approach  When planning a constructivist approach to the mathematical concepts under consideration, the focus is on the left-hand side of Figure 1. On the other hand, when offering students opportunities to construct links between representation forms of the concept, the focus is on the right-hand box, which describes the stages of mathematical concept building.  In learning situations, however, students must have freedom to “jump the gun”.

Interplay between Systematic Approaches and Minimalist Instruction Our experiences of the ClassPad project show (see Eronen & Haapasalo 2006) that this can be realized by organizing different kinds of task types to form a “problem buffet”  To go for linear function, one student team initially picked quite a complicated problem series on optimizing mobile phone costs. After realizing that the (partly linear) cost models appeared too difficult for them, they then chose a new, much easier, problem set, which happened to consist of identification tasks

Learning by Design Eskelinen & Haapasalo (2006) uncover how different kinds of approaches and support for reflective communication affect students’ conceptions of teaching and learning, group dynamics and interest in ICT support.  The developmental approach based on spontaneous procedural knowledge seems to be appropriate concerning as well cognitive as affective variables.  To apply the educational approach to stress the importance of conceptual knowledge, educator needs a lot of sensitivity concerning cognitive and emotional variables in the learning process.

Learning by Design There are numerous researchers who see that Learning by Design is one of the most sophisticated way to implement technology even for young children, opening new productive ways to develop constructively orientated teacher education and service-in-training (e.g. Ojala et al. 1996).  Those who learn more from the instructional materials are their developers, not users.  Therefore teachers and students should design ICT- based lessons and thus become knowledge constructors rather than knowledge users.

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